is reiki a religion?

No. Reiki is strictly a form of energy work, and the principles can be approached through a scientific lens.  Reiki works no matter what the patient or the practitioner's religious beliefs are.  A practitioner can enter a relaxed meditative, prayer-like state while giving Reiki, but the practice itself is not affiliated with any specific religious belief.  Dr. Usui, the founder of Reiki, provided "guidelines" for living a happy life, called the Reiki Precepts, which align with any spiritual practice.